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Two regularization methods for a Cauchy problem for the Laplace equation. (English) Zbl 1132.35493
Summary: A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for $y=0$, and boundary data are prescribed for $x=0$ and $x=\pi $. The solution is sought for $0<y\leqslant 1$. We propose two different regularization methods for the ill-posed problem based on separation of variables. Both methods are applied to find regularized solutions which are stably convergent to the exact one with explicit error estimates.

35R25Improperly posed problems for PDE
35J25Second order elliptic equations, boundary value problems
35B35Stability of solutions of PDE
35A35Theoretical approximation to solutions of PDE
Full Text: DOI
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